Difference of Sigmoidal¶

The Difference of Sigmoidal Membership Functions implements μ(x) = s₁(x) - s₂(x), where each s is a logistic curve with its own slope and center parameters. This creates complex membership shapes by combining two sigmoid functions.

Mathematical Definition¶

The Difference of Sigmoidal Membership Functions is defined as:

$$\mu(x) = s_1(x) - s_2(x)$$

Where each sigmoid function is:

$$s_1(x) = \frac{1}{1 + e^{-a_1(x - c_1)}}$$ $$s_2(x) = \frac{1}{1 + e^{-a_2(x - c_2)}}$$

Parameters¶

The function is characterized by four parameters (two for each sigmoid):

• a₁: Slope parameter for the first sigmoid (s₁)

  • Positive values: standard sigmoid (0 → 1 as x increases)
  • Negative values: inverted sigmoid (1 → 0 as x increases)
  • Larger |a₁|: steeper transition for s₁

• c₁: Center parameter for the first sigmoid (s₁)

  • Controls the inflection point where s₁(c₁) = 0.5
  • Shifts s₁ left/right along the x-axis

• a₂: Slope parameter for the second sigmoid (s₂)

  • Same interpretation as a₁ but for s₂

• c₂: Center parameter for the second sigmoid (s₂)

  • Same interpretation as c₁ but for s₂

Parameter Constraints¶

• a₁ ≠ 0 and a₂ ≠ 0: Cannot be zero (would result in constant functions)
• All parameters can be any real numbers otherwise

Partial Derivatives¶

For optimization in ANFIS networks, we need the gradients of the membership function with respect to each parameter. Since μ(x) = s₁(x) - s₂(x), the derivatives follow from the chain rule:

Derivative w.r.t. Parameters of First Sigmoid (s₁)¶

For s₁(x) = 1/(1 + exp(-a₁(x - c₁))):

• ∂μ/∂a₁ = ∂s₁/∂a₁ = s₁(x) · (1 - s₁(x)) · (x - c₁)
• ∂μ/∂c₁ = ∂s₁/∂c₁ = -a₁ · s₁(x) · (1 - s₁(x))

Derivative w.r.t. Parameters of Second Sigmoid (s₂)¶

For s₂(x) = 1/(1 + exp(-a₂(x - c₂))), and since μ(x) = s₁(x) - s₂(x):

• ∂μ/∂a₂ = -∂s₂/∂a₂ = -s₂(x) · (1 - s₂(x)) · (x - c₂)
• ∂μ/∂c₂ = -∂s₂/∂c₂ = -(-a₂ · s₂(x) · (1 - s₂(x))) = a₂ · s₂(x) · (1 - s₂(x))

Derivative w.r.t. Input (Optional)¶

For chaining in neural networks:

$$\frac{d\mu}{dx} = \frac{ds_1}{dx} - \frac{ds_2}{dx}$$

Where: $$\frac{ds_1}{dx} = a_1 \cdot s_1(x) \cdot (1 - s_1(x))$$ $$\frac{ds_2}{dx} = a_2 \cdot s_2(x) \cdot (1 - s_2(x))$$

Gradient Computation Details¶

The gradients are computed using the fundamental sigmoid derivative property: $$\frac{d}{dx}\left(\frac{1}{1+e^{-z}}\right) = \frac{1}{1+e^{-z}} \cdot \left(1 - \frac{1}{1+e^{-z}}\right) = s(x) \cdot (1 - s(x))$$

This property is used extensively in neural network backpropagation and makes the DiffSigmoidalMF computationally efficient for optimization.

Python Example¶

Let's create a difference of sigmoidal membership functions and visualize its components:

In [5]:

import numpy as np
import matplotlib.pyplot as plt

from anfis_toolbox.membership import DiffSigmoidalMF

diff_sigmoid = DiffSigmoidalMF(a1=2, c1=4, a2=5, c2=8)

x = np.linspace(0, 10, 100)
y = diff_sigmoid(x)

plt.plot(x, y)
plt.show()

import numpy as np import matplotlib.pyplot as plt from anfis_toolbox.membership import DiffSigmoidalMF diff_sigmoid = DiffSigmoidalMF(a1=2, c1=4, a2=5, c2=8) x = np.linspace(0, 10, 100) y = diff_sigmoid(x) plt.plot(x, y) plt.show()

Visualization¶

The following interactive plot shows different difference of sigmoidal membership functions with varying parameter combinations. Each subplot demonstrates how the combination of two sigmoids creates complex membership shapes.